Elliptic Weingarten surfaces in R3 with convex planar boundary

A surface Σ immersed in R3 is an elliptic Weingarten surface if its principal curvatures k1 and k2 satisfy an equation of the type W(k1, k2) = 0, for some function W : R2 → R of class C1 such that (∂W/∂k1) (∂W/∂k2)> 0 on W^{−1}({0}).

Known examples of elliptic Weingarten surfaces include minimal and constant mean curvature surfaces, and surfaces of positive constant gaussian curvature. In 1996 A. Ros and H. Rosenberg proved that for a strictly convex curve Γ ⊂ {z =0} ⊂ R3, there exists a constant h depending only on the curve Γ such that any compact surface embedded in R3^+ := {z ≥ 0} with constant mean curvature H ≤ h must be topologically a closed disk.

In this talk we will present a generalization of Ros-Rosenberg Theorem for elliptic Weingarten surfaces in R3, discussing its proof, which is based on some geometric analysis techniques as the Maximum Principle and the Alexandrov Reflection Method, and the

recent classification of elliptic Weingarten surfaces of revolution obtained by I. Fernandez and P. Mira.

This is a joint work with B. Nelli and G. Pipoli.